In order to accurately predict the power loss generated by a meshing gear
pair the gear loss factor must be properly evaluated. Several gear loss
factor formulations were compared, including the author's approach.

A gear loss factor calculated considering the load distribution along the
path of contact was implemented.

The importance of the gear loss factor in the power loss predictions was put
in evidence comparing the predictions with experimental results. It was
concluded that the gear loss factor is a decisive factor to accurately
predict the power loss. Different formulations proposed in the literature
were compared and it was shown that only few were able to yield satisfactory
correlations with experimental results. The method suggested by the authors
was the one that promoted the most accurate predictions.

Introduction

According to tribology is an important field
in engineering which can contribute to develop more reliable and efficient
mechanisms like gearboxes.

According to the power loss in a gearbox
consists of gear, bearing, seals and auxiliary losses. Gear and bearing
losses can be separated in no-load and load losses. No-load losses occur with
the rotation of mechanical components, even without torque transmission.
No-load losses are mainly related to lubricant viscosity and density as well
as immersion depth of the components on a sump lubricated gearbox, but it
also depends on operating conditions and internal design of the gearbox
casing. Rolling bearing no-load losses depend on type and size, arrangement,
lubricant viscosity and immersion depth.

Load dependent losses occur in the contact of the power transmitting
components. Load losses depended on the transmitted torque, coefficient of
friction and sliding velocity in the contact areas of the components. Load
dependent rolling bearing losses also depend on type and size, rolling and
sliding conditions and lubricant type .

At nominal loads the power loss generated in a gearbox is mainly dependent of
the gears load power losses, which puts in evidence the importance of the
evaluation of the gear loss factor.

This work shows the influence of the gear loss factor formulation
(considering different gear geometries) in the prediction of the power loss.
The gear loss factor formulations will be compared with experimental results
previously published by .

Load dependent power loss in meshing gears

introduced an approach for prediction of the load
dependent losses on spur gears. The power loss generated between gear tooth
contact can be calculated according to Eq. (),
PVZP=PIN⋅HV⋅μmZ.HV is the gear loss factor.

Originally Eq. () was obtained assuming a constant
coefficient of friction (μmZ). This was a simplification of the problem.

Equation () can be used to calculate the average power
loss between gear teeth, given the correct gear loss factor HV. Despite
considering βb the Eq. () initially proposed
by is mostly valid for spur gears .
HVOhl=(1+u)⋅πz1⋅1cos⁡βb⋅1-ϵα+ϵ12+ϵ22
The classical formulas for gear loss factor (Eqs. and )
consider a rigid load distribution, and a constant
coefficient of friction, but tooth profile modifications are disregarded. In
depth details about these formulas can be found in the classical works of
and .

proposed the gear loss factor that is shown in Eq. ().
HVNie=(1+u)⋅πz1⋅1cos⁡βb⋅ϵα⋅1ϵα-1+2k02+2k0+1⋅ϵα

also introduced a Eq. () for
the gear loss factor of a meshing gear pair.
HVBuc=(1+u)⋅πz1⋅1cos⁡βb⋅ϵα⋅2k02-2k0+1
where
k0=z12π⋅ϵα⋅u⋅ra2rp22⋅1cos⁡αt2-112-tan⁡αt

The more recent approach of includes the effects of
profile modifications, keeps the constant coefficient of friction assumption, but no a priori
assumptions about the load distribution are made.

which did no a priori assumption on tooth
load distribution by using generalized displacements, in order to calculate
the efficiency of a meshing gear pair, obtained a closed form solution for
the efficiency of a meshing gear pair (constant coefficient of friction was
assumed) as presented in Eq. (). It turns out that Eq. ()
suggested by Buckingham is an approximation of
the one suggested by when μ≪ 1.
ρ=1-μ⋅(1+u)⋅πz1⋅1cos⁡βb⋅ϵα⋅Λ(μ)
where Λ(μ) is the loss factor described in Eq. ().
Λ(μ)=2k02-2k0+11-μ⋅tan⁡αt⋅2k0-1-πz1⋅ϵα⋅2k02-2k0+1cos⁡βb

The load distribution (force per unit of length along the path of contact)
disregarding elastic effects can be calculated dividing the total normal
force Fn=Mirbi by the total length of the lines of contact
along the path of contact.

The total length of the lines of contact along the path of
contact can be calculated with the algorithm presented in Appendix .
The load distribution per unit of length along the path
of contact can then be calculated according to Eq. ().
An example of the load distribution in a helical gear
is presented (Fig. ).
FN(x,y)=FbnL(x,y)

Load distribution of a helical gear with an applied torque of
320 Nm.

The gear loss factor can now be calculated according to Eq. ()
proposed by HVnum=1pb∫0b∫AEFN(x,y)Fb⋅Vg(x,y)Vbdxdy.
To solve Eqs. () and () the total
length of contacting lines should be known at each point along the path of
contact. To perform this task, an algorithm was developed and implemented
(Appendix ).

Average coefficient of friction

Several authors
have introduced different formulas to calculate the average coefficient of
friction between gear teeth for different gear geometries. Due to the
complexity of the problem, these equations are usually based in experimental
results, and naturally, the results yielded by these models vary for the same
operating conditions. In this work, instead of calculating the coefficient of
friction yielded by these formulations, a value is calculated from the
experimental procedure used in a previous work and then
compared to the models.

Gear loss factor comparisson with different formulas.

Assuming that PVZ0, PVL and PVD are correctly calculated the
power loss generated by the meshing gears can be obtained according to
Eq. (). The rolling bearing, seals and load independent
gear losses were discussed in previous works of .
PVZPexp=PVexp-PVZ0+PVL+PVD

Considering the power loss generated by the gears in the gearbox (Eq. )
an average coefficient of friction (μmZexp)
can be calculated. It can be calculated according to different approaches:

From Ohlendof's approach (Eq. ).μmZexp=PVZPexpPIN⋅HVi

HVi is the gear loss factor which can assume various forms, depending on
the formulation that is used. Four HV were defined according to Eq. ()
HVOhl, Eq. () HVnum,
Eq. HVNie, Eq. HVBuc.

Considering the average power loss generated between gear teeth along the
path of contact according to , μmZexp can be
obtained solving Eq. () to find μmZexp.PVZP=PIN⋅μmZexp⋅(1+u)⋅πz1⋅cos⁡βb⋅ϵα⋅ΛμmZexp

The coefficient of friction extracted from the gear mesh power loss obtained
with Eq. () will be dependent of the formulation that
is used to calculate the gear loss factor. In order to decide which gear loss
factor formulation is better suited for the authors study, this factor was
calculated for seven different gear geometries, in which, spur, helical and
low loss gears are included (Table ) .
The gear loss factor was also calculated based on the results obtained with
the commercial software KissSoft which accounts for elastic effects.

Figure shows the comparison between the different
gear geometries as a function of the k0 (Eq. ) parameter. There are
clearly two groups of results that diverge at a certain point. A deviation is
found in the solutions proposed by ,
and because Eq. () is expected to
yield values between 0 and 0.5. which means that it is not suitable for gears
with profile shift.

The H501 and H951 geometries were previously tested for power loss in an FZG
test rig . The results presented were collected for
FZG load stages with a lever arm of 0.35 m, i.e. K5 = 105, K7 = 199 and K9 = 323 Nm
applied on wheel. Changing from H501 to H951 resulted in a dramatic power
loss reduction (Fig. ), which was attributed to the
H951 gear geometry (everything but the gear geometry was kept the same).
These experimental results suggest that the gear loss factor of the H951 must
be lower than that of the H501. The trends shown by the gear loss factors
obtained with KissSoft, the author's method and Ohlendorf are in
agreement with the experimental observations of . The
gear loss factors obtained with Eq. () are close to those
obtained with the ones derived from the KissSoft computations.
Aiming for simplicity and fast computing the gear loss factor was calculated
using Eq. ().

Following Fig. it becomes clear that Buckingham,
Velex and Winter's approaches are not suitable for all gear geometries and
can only be applied over a limited range of the k0 parameter.

Validation with experimental results

In order to validate the gear loss factor that was proposed, Schlenk's
coefficient of friction was used (Eq. ).
The lubricant parameter (XL) was previously
determined with a spur gear geometry (C40) for different wind turbine gear oil
formulations . Alternatively, experimental results obtained
with H501 and H951 gear geometries were presented in Fig. 3 . The gear loss
factors calculated according to different approaches for the C40, H501 and
H951 gear geometries are presented in Table .
μmZSchlenk=0.048⋅Fbt/bνΣC⋅ρredC0.2⋅η-0.05⋅Ra0.25⋅XL

In Fig. the absolute error of the power loss model
prediction using the KissSoft, Ohlendorf and Author gear loss factors is
presented. The results suggest that the gear loss factor presented by the
authors in Eq. (), considering the rigid load distribution,
present a much lower absolute error for the prediction of a mineral wind turbine
gear oil power loss for with helical gears, previously published by .

Schlenk's Equation should be valid for both helical and spur gear geometries,
also HVOhl is mostly valid for spur gears. This means that using the
lubricant parameter XL extracted from experimental results with spur gears
and applying it to helical gears resulted in excellent correlations between
numerical and experimental data when using HVnum.

Correlation between the experimental power loss measured and the
predicted with Author, Ohlendorf or KissSoft gear loss factors.

Conclusions

In this work several gear loss factors were compared. The gear loss factor
results were indirectly compared with experimental gear power loss
measurements in order to assess the validity of each one of the formulations.

An alternative formulation based on the numerical integration of the rigid
load distribution is suggested. The method presented by the authors to solve
the gear loss factor formula proposed by disregards the
elastic effects of the gears but proved to be reliable to predict the average
power loss of helical and spur gears as proven with experimental results.

The results suggest that the classical formulas are accurate only in very
specific scenarios. The comparison with the experimental results indicates
that the approach suggested by the authors works quite well.

This study has shown the importance of a correct evaluation of the gear loss
factor in the prediction of the power loss generated in meshing gears.

Load distribution along the path of contact

Before enter the contact zone of a gear, or the path of contact which value
is given by Eq. (), a teeth contact line has the
representation of Fig. a.
AE=ϵα⋅pbt

When the contact starts, the length of the contacting line increases
proportionally to the coordinate of the path of contact, represented by the
first condition of Eq. () (Fig. a and b). The contact then continues to increase up to the
situation of a full line of contact, that occur at the coordinate
x=ϵβ⋅pbt=b⋅tan⁡βb up to the end of
contact at x=ϵα⋅pbt which is given by second
condition of Eq. () (Fig. c). Then,
the teeth start to go out from the contact and the line length starts to
decrease as shown in the third condition of Eq. () and Fig. d.
l(x)=xsin⁡βb0<x<ϵβ⋅pbtbcos⁡βbϵβ⋅pbt<x<ϵα⋅pbtbcos⁡βb-x-ϵα⋅pbtsin⁡βbϵα⋅pbt<x<ϵα+ϵβ⋅pbt
Equation () previously presented is valid for the length
of a single line along the path of contact. The other teeth have the same
behaviour of the single line yet presented but at the distance of a
transverse pitch (pbt), which is the distance between the teeth along the path of
contact as represented in Fig. .

Evolution of a single line along the path of contact.

The same equations deduced for a single line can be used, but the coordinates
should be transformed according to Eq. (). The value i
of the Eq. () is calculated with Eq. ()
that represents the lines screened from the single line
with value i= 0, from behind and behead in integer steps.
x*(x)=x+i⋅pbti=-ceilϵα+ϵβ:1:ceilϵα+ϵβ
Ceil is a function that rounds the value for the highest close integer.

It is also possible to do a 3-D representation of the line length as function
of x and y. To do that, the y coordinate representing the tooth width
that changes from 0 up to b. Since the tooth line of contact of a helical
gear has a helix angle the y coordinate is function of the x coordinate
which can be expressed with Eq. ().
x(x,y)*=x+i⋅pbt+ytan⁡βb
Applying the coordinate transformation of Eq. () and
the formulas of Eq. (), the line length of each tooth
screened from the teeth i is presented in Eq. ().
li(x,y)=x*sin⁡βb0<x*<ϵβ⋅pbtbcos⁡βbϵβ⋅pbt<x*<ϵα⋅pbtbcos⁡βb-x*-ϵα⋅pbtsin⁡βbϵα⋅pbt<x*<ϵα+ϵβ⋅pbt

The formulation presented is valid for gears with a contact ratio
ϵα>ϵβ.

For the case that one complete line is not in contact, the cycle of meshing
is slightly different and the path of contact is smaller than the transverse
pitch. In such cases usually the overlap contact ratio is
ϵβ>ϵα.

The equation is slightly different from that presented before because the
domains change in a different way as presented in Eq. ().
li(x,y)=x*sin⁡βb0<x*<ϵα⋅pbtϵα⋅pbtsin⁡βbϵα⋅pbt<x*<ϵβ⋅pbtϵα⋅pbtsin⁡βb-x*-ϵβ⋅pbtsin⁡βbϵβ⋅pbt<x*<ϵα+ϵβ⋅pbt
The total sum of lines can be easily done with Eq. ().
It is important to note that the algorithm also calculate the line contact
length of spur gears using only the second row of Eq. ().
L(x,y)=∑-ceilϵα+ϵβceilϵα+ϵβli(x,y)

Stepwise functions

The algorithm previously presented is based on the identification of
different domains in the meshing cycle of helical gears. However, the
different domains can be combined using stepwise functions like Heaviside
(Eq. ) or hyperbolic tangent (Eq. ).
ξ=11+e-2k(x-a)ξ=12⋅(tanh⁡(k⋅(x-a))+1)
The coordinate a is the point when the step is desired.

Using the hyperbolic tangent equation, the three domains can be expressed in
Eq. () for the beginning of contact (a= 0),
Eq. () for a complete line (a=ϵβ⋅pbt)
and Eq. () for a line going out from the contact
(a=ϵα⋅pbt). The constant k changes the precision
of the algorithm. For the case it was considered k= 1000.
ξ1=12⋅tanh⁡(k⋅x)-tanh⁡k⋅x-ϵα+ϵβ⋅pbtξ2=12⋅tanh⁡k⋅x-ϵβ⋅pbt+1ξ3=12⋅tanh⁡k⋅x-ϵα⋅pbt+1
For each single line the length along the path of contact is given by Eq. ().
l(x)=1sin⁡βb⋅ξ1⋅x-ξ2⋅x-ϵβ⋅pbt-ξ3⋅x-ϵα⋅pbt
For spur gears the length for each line is given by Eq. ().
l(x)=b⋅ξ1

For the lines screened from the one considered the length is computed with
Eq. () previously explained which results in Eq. ().
li(x,y)=lx(x,y)*
The total sum of lines is then given by Eq. ().

Using such type of function or other stepwise function is great to get a
continuous function. However, the computational time can increase due to the
expense of computing the step function. The algorithm with step function
works for all the type of gear geometries and the transverse and overlap
contact ratios (ϵα and ϵβ) do not need to follow any rule.